[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207808-en":3,"doc-seo-207808-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207808,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",4,"Exam","Matrices - Mathematics Notes - Unit 3","Matrices focuses on the key foundations needed for working with 2×2 and 3×3 matrices. It introduces essential definitions, including matrix order, entries aij, the leading diagonal, and special matrices such as the identity and zero matrices. The notes then build core algebra skills: addition, subtraction, scalar multiplication, and matrix multiplication, emphasizing valid dimensions. Applications connect matrices to solving systems of equations, geometric transformations, chemical balancing, circuit currents, and fundamental physical laws.","Matrices  \nPrerequisites: Adding, subtracting, multiplying and dividing numbers;  \nelementary row operations.  \nMaths Applications: Solving systems of equations; describing geometric  \ntransformations; deriving addition formulae.  \nReal-World Applications: Balancing chemical equations; flight  \nstopover information; currents in electrical circuits; formulation of fundamental physical laws.  \nBasic Definitions  \nDefinition:  \nA matrix is a rectangular array of numbers (aka entries or elements) in parentheses, each entry being in a particular row and column.  \nDefinition:  \nThe order of a matrix is given as m × n (read m by n), where m is the number of rows and n the number of columns and is written as,  \n􀀁􀀃 a11 a12 a13 … a1n 􀀂􀀄  \ndef 􀀃 a21 a22 a23 … a2n 􀀄 A ≡ (aij ) m×n = 􀀃 a31 a32 a33 … a3n 􀀄  \n􀀃 􀀂 􀀂 􀀂 􀀃 􀀄  \n􀀅􀀃 am1 am2 am3 amn 􀀆􀀄  \nThe element in row i and column j of a matrix is written as aij and called the (i, j) th entry of A.  \nIn this course, we will deal almost exclusively with matrices that have orders 2 × 2 and 3 × 3.  \nM Patel (April 2012) 1 St. Machar Academy  \nAdvanced Higher Notes (Unit 3) Matrices  \nDefinition:  \nThe main diagonal (aka leading diagonal) of any matrix is the set of entries aij where i = j.  \nSpecial cases arise when either m = 1 or n = 1.  \nDefinition:  \nA row matrix is a 1 × n matrix and is written as,  \n(a11 a12 … a1(n −1) a1n )  \nA column matrix is a m × 1 matrix and is written as,  \n􀀁􀀃 a11 􀀂􀀄  \n􀀃 a21 􀀄  \n􀀃 􀀂 􀀄  \n􀀃 a( m −1)1 􀀄  \n􀀃􀀅 am1 􀀆􀀄  \nThe case when m = n is a very important one.  \n\n| Definition:\u003Cbr>A square matrix (of order m × m) is a matrix with the same number of rows as columns (equal to m) and is written as, |  |  |\n| --- | --- | --- |\n| 􀀁\u003Cbr>􀀃\u003Cbr>􀀃\u003Cbr>􀀃\u003Cbr>􀀃􀀅 | a11 a12 … a1m\u003Cbr>a21 a22 … a2m 􀀂 􀀂 􀀃 am1 am2 amm | 􀀂\u003Cbr>􀀄\u003Cbr>􀀄\u003Cbr>􀀄\u003Cbr>􀀆􀀄 |\n\nDefinition:  \nThe identity matrix (of order m) is the m × m matrix all of whose entries are 0 apart from those on the main diagonal, where they all equal 1,  \nM Patel (April 2012) 2 St. Machar Academy  \nAdvanced Higher Notes (Unit 3) Matrices  \n􀀁 1 0 … 0􀀂  \n􀀃 􀀄  \nIm  􀀃 0 1 … 0 􀀄 􀀅0􀀂 0􀀂 􀀃 1 􀀆  \nDefinition:  \nThe zero matrix (of order m × n) is the m × n matrix all of whose entries are 0,  \n􀀁 0 0 0 … 0􀀂  \n􀀃 􀀄  \ndef 􀀃 0 0 0 … 0 􀀄  \nOm × n = 􀀃 0 0 0 … 0 􀀄􀀃 􀀂 􀀂 􀀂 􀀃 􀀄  \n􀀅􀀃 0 0 0 0 􀀆􀀄  \nFor a square zero matrix, sometimes the notation Om is used.  \nMatrix Algebra  \nAddition, Subtraction and Scalar Multiplication  \nDefinition:  \nThe matrix sum of A and B is obtained by adding corresponding entries of A and B,  \ndef  \n(a + b)ij = aij + bij  \nThe matrix difference of A and B is obtained by subtracting the entries of B from the corresponding ones in A,  \ndef  \n(a − b)ij = aij − bij  \nNote that matrix addition and subtraction only makes sense if A and B both have the same order.  \nM Patel (April 2012) 3 St. Machar Academy  \nAdvanced Higher Notes (Unit 3) Matrices Example 1  \n􀀁 0 −3􀀂 􀀁 13 1 􀀂  \nAdd the matrices A = 􀀃 􀀄 and B = 􀀃 􀀄 .  \n􀀅5 2 􀀆 􀀅−4 5 􀀆  \nAs the matrices have the same order, they can be added.  \nA + B = 􀀁􀀃 0 −3􀀂􀀄 + 􀀁􀀃 13 1 􀀂􀀄  \n􀀅5 2 􀀆 􀀅−4 5 􀀆  \n􀀁 0 + 13 −3 + 1􀀂  \n= 􀀃 􀀄  \n􀀅5 + (−4) 2 + 5 􀀆  \n􀀁13 −2􀀂= 􀀃 􀀄  \n􀀅1 7 􀀆  \nExample 2  \nFind the difference P − Q where P = 􀀁􀀃 4 −2 0 􀀂􀀄 and  \n􀀅1 0 37 􀀆  \n􀀁 −4 −2 2􀀂  \nQ = 􀀃 􀀄  \n􀀅−8 3 6 􀀆 .  \nP − Q = 􀀁􀀃 4 −2 0 􀀂􀀄 − 􀀁􀀃 −4 −2 2􀀂􀀄  \n􀀅1 0 37 􀀆 􀀅−8 3 6 􀀆  \n􀀁 8 0 −2􀀂  \n= 􀀃 􀀄  \n􀀅9 −3 31 􀀆  \nDefinition:  \nThe scalar multiplication of A by k (k ∈ 􀀄 ) is obtained by multiplying each entry of A by k,  \ndef  \n(ka )ij = kaij  \nM Patel (April 2012) 4 St. Machar Academy  \nAdvanced Higher Notes (Unit 3) Matrices Example 3  \n􀀁 4 −4 0 􀀂  \n􀀃 􀀄    \nIf A = 􀀃 5 6 1 􀀄 , calculate 12 A.  \n􀀃􀀅8 9 −1 􀀆􀀄  \n􀀁 4 −4 0 􀀂     􀀃 􀀄  \n12 A = 12 􀀃 5 6 1 􀀄􀀃􀀅8 9 −1 􀀆􀀄  \n􀀁 4 / 2  \n􀀃= 􀀃 5 / 2  \n􀀃􀀅8 / 2  \n−4 / 2 6 / 2  \n9 / 2  \n0 / 2 􀀂  \n􀀄  \n1 / 2 􀀄−1 / 2 􀀆􀀄  \n􀀁 2 −2  \n􀀃  = 􀀃 52 3  \n􀀃􀀅4 ~~ 9~~2  \n0 􀀂   􀀄  \n12 􀀄  1 􀀄  \n− 2 􀀆  \nMatrix Multiplication  \nDefinition:  \nThe matrix product of A and B, where A is of order m × n and B is of order n × p is obtained b","cbCait6IAB7yn1oD","https://ap.wps.com/l/cbCait6IAB7yn1oD","pdf",218147,1,32,"English","en",105,"# Basic Definitions\n## Matrix order and entries\n## Main diagonal, row and column matrices\n## Square, identity, and zero matrices\n# Matrix Algebra\n## Addition, subtraction, and scalar multiplication\n## Matrix multiplication\n## Transpose and powers","[{\"question\":\"What does the order of a matrix mean?\",\"answer\":\"The order of a matrix is written as m×n, where m is the number of rows and n is the number of columns.\"},{\"question\":\"When can two matrices be added or subtracted?\",\"answer\":\"Matrix addition and subtraction only make sense when both matrices have the same order.\"},{\"question\":\"How is matrix multiplication AB defined?\",\"answer\":\"If A is m×n and B is n×p, the product AB has entries (AB)ij found by taking the scalar product of the i-th row of A with the j-th column of B.\"}]","Matrices - Mathematics Notes - Unit 3 | 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